10/23/2021
1.- Executive Summary.
2.- Regression AAPL vs S&P500.
3.- AAPL Close vs SP500 & NASDAQ100.
4.- Months, days and trading sessions.
5.- Regression AAPLre vs SP500re.
6.- Betas daily rolling 3,5,7,10 years.
7.- Beta summary.
8.- Look ahead: Building a Forecast model.
9.- Model Structure.
10.-Regression Model: dataset.
11.-Time-related predictors.
12.-Market-related predictors.
13.-Discount rates -related predictors.
14.-Model Output: ANOVA.
15.-Model Output: Coefficients.
16.-But there is a concern: multicolinearity.
17.-Predictors correlations.
18.-Residuals Plot.
19.-Root Mean Squared Error (RMSE).
20.-Apple Close (Forecast).
21.-Risk: Probability of permanent loss.
22.-Risk goes down with longer hold.
This presentation summarizes the work done to find a model to represent and predict Apple’s share price (ticker AAPL) at the closing of every session, and forecast future prices a number of sessions forward ( in this case 21 sessions, about a month of trading).
The data set covers trading sessions since January 1990 until October 2021. The model is built for the dataset to update itself (through R code reaching out to the Yahoo platform), predict and update the model with new data, and forecast 21 sessions forward.
The last update for the purpose of this document was performed on October 21, 2021.
The closing of AAPL is represented by a linear regression model which includes predictors such us sequential days of trading, year, month, day, the closing level of the S&P500 and the NASDAQ100, 5 year, 10 year, and 30 year Treasury prices, and the dollar index.
The relationship between the predictors and the AAPL shares is not always linear. The predictors are hence transformed to reflect this, and then lagged 21 periods (trading sessions) to forecast forward closings.
There are signs of colinearity (predictors highly correlated with each other) in the model. This needs further research, certainly constitutes an opportunity for improvement, and is beyond the scope of this work for the time being.
The model predicts AAPL closing with an error of 1.40 usd/share on the training set and 10.34 usd/share on the testing set. A total of 87% of the actual data points on the testing set are within 1 standard error of 10.34 usd/share.
Apple’s closing on October 21 was 149.48 usd/share (Yahoo Finance). The model predicts 151.89 usd/share for that day.
The model forecasts AAPL to close on October 22, 2021 (next trading session) at 156.19 usd/share. AAPL actually closed at 148.69 (Yahoo Finance).
The forecast for 25 October was 156.36 usd/share, AAPL closed at 148.64 (Yahoo Finance).
All this may be explored on the graphs. Beta coefficients versus the S&P500 are also derived and explored. The 5 years rolling Beta to the S&P500 reported by the model is 1.21 (verus 1.23 reported by Yahoo Finance). Beta is often interpreted as a measure of volatility and risk. The evolution of Beta for 3,5,7, and 10 years may be explore interactively on the graphs.
This work also suggest a different approach to see and measure risk. I suggest that risk be understood as the probability of permanent loss. That requires a buy-hold-sell strategy within a window of time. The time window goes from 1,5,10,20,50,100,200(approx 1 Year),400,1000,2000(10 Years),4000 trading sessions. The wider the time window, the lower the probability of permanent loss –the lower the risk–.
lm(formula = AAPLClose ~ (. - date - Month - X.TYXClose - DX.Y.NYBClose -
X.DJIClose) + factor(Month, exclude = c(5, 11, 4, 12, 6,
7, 3, 2)), data = FullDF)
[1] "Residuals:"
Min. 1st Qu. Median Mean 3rd Qu. Max. -22.8786 -2.1529 -0.2152 0.0000 3.3252 14.5205
[1] "Adjusted R squared:"
[1] 0.9746672
Adjusted R squred at 0.97 means that the model explains 97.47 percent of the total variance.
[1] "F-statistic:"
value numdf dendf 10442.96 10.00 2704.00
[1] "P-value:"
value
0
High F-statictic and low (zero) p-values means that the model is significant. Perhaps colinearity is present.
Estimate Std. Error t value Pr(>|t|) (Intercept) -7.559765e+05 2.777576e+04 -27.217129 2.244391e-144 id -1.510236e+00 5.549118e-02 -27.215785 2.309875e-144 Year 3.799028e+02 1.395800e+01 27.217575 2.223078e-144 Day 1.050600e+00 4.034714e-02 26.039021 1.442891e-133 SP500 -2.338573e-02 9.150650e-04 -25.556364 3.200437e-129 NDXClose 1.572176e-02 1.923123e-04 81.751207 0.000000e+00 X.TNXClose -2.670087e+00 2.112474e-01 -12.639619 1.272616e-35 X.IRXClose -1.289807e-01 1.159453e-01 -1.112427 2.660533e-01 Month8 2.226430e+02 8.163735e+00 27.272197 6.905232e-145 Month9 2.542560e+02 9.334650e+00 27.237871 1.439944e-144 Month10 2.865822e+02 1.051457e+01 27.255718 9.827244e-145
P-values are all close to zero. This means the coefficient for the predictors are significant; perhaps colinearity is present.
Analysis of Variance Table
Response: AAPLClose
Df Sum Sq Mean Sq F value Pr(>F)
id 1 1328284 1328284 53545.136 < 2.2e-16 ***
Year 1 507 507 20.458 6.358e-06 ***
Day 1 146 146 5.883 0.01535 *
SP500 1 939189 939189 37860.123 < 2.2e-16 ***
NDXClose 1 299840 299840 12086.987 < 2.2e-16 ***
X.TNXClose 1 2292 2292 92.391 < 2.2e-16 ***
X.IRXClose 1 1844 1844 74.339 < 2.2e-16 ***
Month(f) 3 18464 6155 248.105 < 2.2e-16 ***
Residuals 2704 67078 25
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
AAPL Close vs SP500 & NASDAQ100 (y secondary) ( secondary y axis does not display properly in ggplotly. Multiply main y axis by 100 to read SP500 and Nasdaq100.)
Before we look at monthly returns, let’s explore what “Monthly” means as an average long-run number of sessions in the market. The data frame has a total of:
[1] 8015
records. These are daily sessions of the S&P500. These sessions cover dates from:
[1] "1990-01-02"
excluding all Saturdays, all Sundays, and all trading holidays, until:
[1] "2021-10-21"
The difference in months is:
[1] 381.6247
The average length of a month in this data set is:
[1] 21
AAPL Monthly %Returns vs Monthly %Returns SP500 & NASDAQ100
Call:
glm(formula = AAPLre ~ SP500re, data = returnsMonthlyDF)
Deviance Residuals:
Min 1Q Median 3Q Max
-65.152 -6.215 -0.111 6.054 105.809
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.49610 0.12935 11.57 <2e-16 ***
SP500re 1.23440 0.02854 43.26 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for gaussian family taken to be 130.1822)
Null deviance: 1286723 on 8014 degrees of freedom
Residual deviance: 1043150 on 8013 degrees of freedom
AIC: 61774
Number of Fisher Scoring iterations: 2
Beta APPLre vs SP500re:
[1] 1.234399
Yahoo Finance Beta (5Y Monthly) = 1.22
Apple Beta vs SP500 in 3,5,7,10 daily rolling years:
xids beta3Y beta5Y beta7Y beta10Y 346 346 1.25 1.21 1.26 1.21
AAPL Betas vs S&P500 daily rolling 3, 5, 7, 10 Years.
Betas where calculated using rolling:
[1] 21
days. Each iteration of the linear regression is calculated starting
[1] 21
days ahead of the previous one. Betas calculated over different periods of time have fluctuated in the past. They have all converged in recent periods to a range between 1.21, and 1.25. Yahoo finance reports on 2021-10-14 a 5Y Beta for AAPL of 1.23. Pretty good results !
What is APPL Closing price likely to be X trading sessions ahead ?
The model takes 3 types of predictors:
1.- time-related,
2.- market related, and
3.- fundamentals-related.
Each group of predictors aims at explaining different forces which make APPL shares go up and down in price.
Lag: market and fundamentals predictors will be lagged
[1] 21
periods to be able to forecast that same amount periods forward, in the last stage. Lagging variables allows us to bring past data into the future. Time variables do not need to be lagged since sequential session id, year, month and day can be precisely established going forward.
## 'data.frame': 8015 obs. of 10 variables: ## $ id : int 1 2 3 4 5 6 7 8 9 10 ... ## $ date : Date, format: "1990-01-02" "1990-01-03" ... ## $ Year : num 1990 1990 1990 1990 1990 1990 1990 1990 1990 1990 ... ## $ Month : num 1 1 1 1 1 1 1 1 1 1 ... ## $ Day : num 2 3 4 5 8 9 10 11 12 15 ... ## $ YearSq : num 3960100 3960100 3960100 3960100 3960100 ... ## $ MonthSq: num 1 1 1 1 1 1 1 1 1 1 ... ## $ DaySq : num 4 9 16 25 64 81 100 121 144 225 ... ## $ SP500 : num 360 359 356 352 354 ... ## $ SP500Sq: num 129377 128709 126501 124045 125167 ...
## 'data.frame': 8015 obs. of 10 variables: ## $ DX.Y.NYBClose: num 94.3 94.4 92.5 92.8 92.1 ... ## $ NDXClose : num 228 226 225 223 224 ... ## $ X.DJIClose : num 0 0 0 0 0 0 0 0 0 0 ... ## $ X.TNXClose : num 7.94 7.99 7.98 7.99 8.02 8.02 8.03 8.04 8.1 8.1 ... ## $ TNX10Ytrans : num 0.318 0.316 0.316 0.316 0.314 ... ## $ X.IRXClose : num 7.58 7.63 7.59 7.54 7.54 7.55 7.5 7.55 7.5 7.5 ... ## $ TIX5Ytrans : num 0.334 0.332 0.334 0.336 0.336 ... ## $ X.TYXClose : num 8 8.04 8.04 8.06 8.09 8.1 8.11 8.11 8.17 8.17 ... ## $ TYX30Ytrans : num 0.315 0.313 0.313 0.313 0.311 ... ## $ AAPLClose : num 0.333 0.335 0.336 0.337 0.339 ...
These predictors explain the overall trend overtime.
id = sequential number of sessions since January 1990.
date = chr string (Not used as a predictor).
Year = numeric value of the year.
YearSq = the square of Year. (Not significant. Not used in the model.)
Month = numeric value of the month.
MonthSq = the square of month.
Day = numeric value of the day.
DaySq = the square of day. (Not significant. Not used in the model.)
These predictors explain the overall market forces of supply and demand.
SP500 = the closing of the index for each session.
SP500Sq = the square of SP500.
DX.Y.NYBClose = dollar index close for each session.
NDXClose = the closing of NASDAQ100 index for each session.
NDXCloseSq = the square of NDXClose. (Not significant. Not used in the model.)
X.DJIClose = the closing of Dow Jones index for each session. (Not significant. Not used in the model.)
These predictors are connected to the fundamental valuation of the stock through discounted cashflow.
X.TNXClose = 10-year treasury bill close.
TNX10Ytrans = transformation of TNXClose.
X.IRXClose = 5-year treasury bill close.
TIX5Ytrans = transformation of IRXClose.
X.TYXClose = 30-year treasury bill close.
TYX30Ytrans = transformation of TYXClose.
Analysis of Variance Table
Response: AAPLClose
Df Sum Sq Mean Sq F value Pr(>F)
id 1 44094 44094 22295.518 < 2.2e-16 ***
Year 1 89 89 45.161 1.995e-11 ***
Month 1 445 445 224.981 < 2.2e-16 ***
Day 1 3654 3654 1847.383 < 2.2e-16 ***
MonthSq 1 71 71 35.714 2.426e-09 ***
SP500 1 1980 1980 1000.938 < 2.2e-16 ***
SP500Sq 1 11101 11101 5613.227 < 2.2e-16 ***
DX.Y.NYBClose 1 625 625 316.083 < 2.2e-16 ***
X.TNXClose 1 763 763 385.654 < 2.2e-16 ***
TNX10Ytrans 1 8447 8447 4271.361 < 2.2e-16 ***
X.IRXClose 1 1043 1043 527.540 < 2.2e-16 ***
TIX5Ytrans 1 174 174 87.813 < 2.2e-16 ***
X.TYXClose 1 1955 1955 988.306 < 2.2e-16 ***
TYX30Ytrans 1 650 650 328.585 < 2.2e-16 ***
Residuals 5595 11065 2
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Call:
aov(formula = LagLM)
Terms:
id Year Month Day MonthSq SP500 SP500Sq
Sum of Squares 44093.56 89.31 444.94 3653.54 70.63 1979.54 11101.21
Deg. of Freedom 1 1 1 1 1 1 1
DX.Y.NYBClose X.TNXClose TNX10Ytrans X.IRXClose TIX5Ytrans
Sum of Squares 625.11 762.70 8447.42 1043.31 173.67
Deg. of Freedom 1 1 1 1 1
X.TYXClose TYX30Ytrans Residuals
Sum of Squares 1954.56 649.84 11065.16
Deg. of Freedom 1 1 5595
Residual standard error: 1.406303
Estimated effects may be unbalanced
(Intercept) id Year Month Day
1.812747e+05 3.634371e-01 -9.110017e+01 -7.648393e+00 -2.491013e-01
MonthSq SP500 SP500Sq DX.Y.NYBClose X.TNXClose
-3.202201e-03 -2.240315e-02 1.179981e-05 -5.875717e-02 1.095126e+01
TNX10Ytrans X.IRXClose TIX5Ytrans X.TYXClose TYX30Ytrans
2.101148e+02 -4.447771e-01 -5.809950e+00 -7.543895e+00 -1.730121e+02
The Adjusted R-squared value for the model is:
[1] 0.8712447
which means that 87% of the variance is explained by the model, and suggests good fit.
And its F-statistic is:
value numdf dendf 2712.019 14.000 5595.000
with its corresponding p-value:
value
0
This means that the overall regression model is significant. All the predictor variables are statistically significant at 0.05 significance level.
Let’s look at the variance inflation factor. Do we have multicolinearity?
1st approach: use VIF function from “regclass” package. If VIF is greater than 5, that predictor may be colinear with another.
library( regclass ) ; VIF( LagLM). This library fails to load. I will use the vif() function in the car library instead:
id Year Month Day MonthSq
1.569258e+06 1.566892e+06 3.123737e+03 2.290945e+01 1.344052e+00
SP500 SP500Sq DX.Y.NYBClose X.TNXClose TNX10Ytrans
1.763730e+02 1.138285e+02 2.494302e+00 2.042149e+03 1.441421e+03
X.IRXClose TIX5Ytrans X.TYXClose TYX30Ytrans
1.349987e+02 1.471874e+02 2.615983e+03 1.843824e+03
2nd approach: look at correlations. We want low correlation amongst predictors.
id Year Month Day id 1.000000000 0.998989925 0.0077156840 1.985499e-03 Year 0.998989925 1.000000000 -0.0370486753 -1.748587e-03 Month 0.007715684 -0.037048675 1.0000000000 -2.605396e-04 Day 0.001985499 -0.001748587 -0.0002605396 1.000000e+00 MonthSq 0.024972918 0.006132963 0.4141369438 2.188639e-02 SP500 0.789552387 0.788946731 -0.0089078591 9.124913e-04 SP500Sq 0.738907239 0.738744947 -0.0166544350 6.338672e-04 DX.Y.NYBClose -0.314863992 -0.314734700 -0.0131790951 1.124367e-03 X.TNXClose -0.919088299 -0.918266662 -0.0033253147 1.915890e-04 TNX10Ytrans 0.914464034 0.913807145 0.0018578711 -6.132122e-05 X.IRXClose -0.720296780 -0.720161753 0.0017583367 1.409721e-03 TIX5Ytrans 0.730545283 0.730642890 -0.0058155332 -1.811083e-03 X.TYXClose -0.937447547 -0.936778220 0.0026347476 -1.093113e-03 TYX30Ytrans 0.939591304 0.939110419 -0.0048986340 1.417055e-03
MonthSq SP500 SP500Sq DX.Y.NYBClose id 0.024972918 0.7895523871 0.7389072390 -0.314863992 Year 0.006132963 0.7889467309 0.7387449467 -0.314734700 Month 0.414136944 -0.0089078591 -0.0166544350 -0.013179095 Day 0.021886394 0.0009124913 0.0006338672 0.001124367 MonthSq 1.000000000 0.0050352353 -0.0027088429 -0.036698219 SP500 0.005035235 1.0000000000 0.9869994605 0.065229450 SP500Sq -0.002708843 0.9869994605 1.0000000000 0.045148749 DX.Y.NYBClose -0.036698219 0.0652294500 0.0451487493 1.000000000 X.TNXClose -0.057415666 -0.6726805474 -0.5973264730 0.242118977 TNX10Ytrans 0.057541898 0.6216374433 0.5513162352 -0.298003724 X.IRXClose -0.026769580 -0.3133921818 -0.2318847190 0.279534573 TIX5Ytrans 0.023094762 0.2956558759 0.2175146625 -0.314183879 X.TYXClose -0.057703919 -0.7604060524 -0.6911395610 0.196275631 TYX30Ytrans 0.060995455 0.7212262477 0.6564104529 -0.248829897
X.TNXClose TNX10Ytrans X.IRXClose TIX5Ytrans id -0.919088299 9.144640e-01 -0.720296780 0.730545283 Year -0.918266662 9.138071e-01 -0.720161753 0.730642890 Month -0.003325315 1.857871e-03 0.001758337 -0.005815533 Day 0.000191589 -6.132122e-05 0.001409721 -0.001811083 MonthSq -0.057415666 5.754190e-02 -0.026769580 0.023094762 SP500 -0.672680547 6.216374e-01 -0.313392182 0.295655876 SP500Sq -0.597326473 5.513162e-01 -0.231884719 0.217514663 DX.Y.NYBClose 0.242118977 -2.980037e-01 0.279534573 -0.314183879 X.TNXClose 1.000000000 -9.895185e-01 0.835706217 -0.826740921 TNX10Ytrans -0.989518492 1.000000e+00 -0.844866003 0.851106737 X.IRXClose 0.835706217 -8.448660e-01 1.000000000 -0.989073730 TIX5Ytrans -0.826740921 8.511067e-01 -0.989073730 1.000000000 X.TYXClose 0.981459267 -9.621210e-01 0.738783624 -0.728055889 TYX30Ytrans -0.979667685 9.769873e-01 -0.750334478 0.750438633
X.TYXClose TYX30Ytrans id -0.937447547 0.939591304 Year -0.936778220 0.939110419 Month 0.002634748 -0.004898634 Day -0.001093113 0.001417055 MonthSq -0.057703919 0.060995455 SP500 -0.760406052 0.721226248 SP500Sq -0.691139561 0.656410453 DX.Y.NYBClose 0.196275631 -0.248829897 X.TNXClose 0.981459267 -0.979667685 TNX10Ytrans -0.962120954 0.976987268 X.IRXClose 0.738783624 -0.750334478 TIX5Ytrans -0.728055889 0.750438633 X.TYXClose 1.000000000 -0.992767062 TYX30Ytrans -0.992767062 1.000000000
The correlation matrix shows that there is indeed high correlation between predictors, and the possibility of multicolinearity is real. Based on its Adjusted R squared and p-values, I am taking the model as good, and will not fix colinearity in this iteration of the exercise.
AAPL Close Actual & Regression (Training set).
AAPL Close Actual & Regression (Testing set).
The RMSE for the training set is:
[1] 1.404
and on the testing set is:
[1] 10.35
There is
[1] 87.08
percent of data points Within the black bands (1 standard error form the estimated regression line).
c.1.nfwd. OutOfSample.date AAPLCloseOutofSample 8016 1 2021-10-22 156.1928 8017 2 2021-10-25 156.3612 8018 3 2021-10-26 155.1946 8019 4 2021-10-27 148.0768 8020 5 2021-10-28 148.8331 8021 6 2021-10-29 144.9924 8022 7 2021-11-01 148.9005 8023 8 2021-11-02 144.3647 8024 9 2021-11-03 147.9785 8025 10 2021-11-04 149.3354
c.1.nfwd. OutOfSample.date AAPLCloseOutofSample 8026 11 2021-11-05 152.3610 8027 12 2021-11-08 151.1685 8028 13 2021-11-09 148.7522 8029 14 2021-11-10 147.8090 8030 15 2021-11-11 148.7072 8031 16 2021-11-12 155.1243 8032 17 2021-11-15 156.9915 8033 18 2021-11-16 157.8728 8034 19 2021-11-17 160.8905 8035 20 2021-11-18 162.6581 8036 21 2021-11-19 163.5685
[1] "Out-of-Sample Set. Trading sessions since 2021-10-21"
Apple Close (Forecast) 21 Sessions.
Buy, hold and sell within a window of ‘x’ days. What is the probability of negative returns?
Hold 1 days. Mean= 0.11 SD= 2.75 Prob or perm loss= 48.4 %.
Hold 5 days. Mean= 0.57 SD= 6.05 Prob or perm loss= 46.25 %.
Hold 10 days. Mean= 1.15 SD= 8.61 Prob or perm loss= 44.69 %.
Hold 20 days. Mean= 2.34 SD= 12.35 Prob or perm loss= 42.49 %.
Hold 50 days. Mean= 6.05 SD= 20.31 Prob or perm loss= 38.29 %.
Hold 100 days. Mean= 12.43 SD= 31.1 Prob or perm loss= 34.47 %.
Hold 200 days. Mean= 26 SD= 48.72 Prob or perm loss= 29.68 %.
Hold 400 days. Mean= 59.45 SD= 93.32 Prob or perm loss= 26.2 %.
Hold 1000 days. Mean= 208.04 SD= 306.1 Prob or perm loss= 24.84 %.
Hold 2000 days. Mean= 833.55 SD= 1075.84 Prob or perm loss= 21.92 %.
Hold 4000 days. Mean= 6977.61 SD= 5358.08 Prob or perm loss= 9.64 %.
DaysHold ProbOfLoss 1 1 48.40 2 5 46.25 3 10 44.69 4 20 42.49 5 50 38.29 6 100 34.47 7 200 29.68 8 400 26.20 9 1000 24.84 10 2000 21.92 11 4000 9.64
For the computation of probabilities, the distribution is assumed to be normal. But that is not always true! See the histograms for a better feel of where losses may take place. Holding times up to 20 days (approx 1 month) carry about 50% chance of permanent loss. Above 50 days (approx 2 months) the probability of permanent loss diminishes significantly and stabilizes around the 25% mark for 1 year and above.